DOCSLIB.ORG

Numerical Analysis in Nonlinear Least Squares Methods and Applications

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Numerical Analysis in Nonlinear Least Squares Methods and Applications Department of Electrical and Computer Engineering Numerical Analysis in Nonlinear Least Squares Methods and Applications Christina Nguk Ling Eu This thesis is presented for the Degree of Master of Philosophy (Electrical and Computer Engineering) of Curtin University December 2017 DECLARATION To the best of my knowledge and belief this thesis contains no material previously published by any other person except where due acknowledgment has been made. This thesis contains no material which has been accepted for the award of any other degree or diploma in any university. i ACKNOWLEDGEMENTS I would like to express my special thanks to my main supervisor, Professor Bean San Goh, for his quality guidance, continual encouragement and endless support through- out my MPhil study. During my research, he has provided mewith invaluable academic advice and insightful suggestions and steered me into the right direction whenever I am heading towards the wrong path in my research. The door to his office was always open whenever I ran into a trouble spot or had a problem on my research or writing. He is not only a good supervisor but as a good friend as well. I would also like to thank my co-supervisor, Dr. Hendra Gunawan Harno, for his continual patience, guidance and support throughout my research. He has always been helpful and supportive whenever I faced any academic and administrative issues. He always tried his best to help me by providing any possible assistance. My sincere thanks also goes to my second co-supervisor, Dr. Garenth King Hann Lim, for his support and cooperation throughout my years of study. Without their passionate input, this research project could not have been successfully conducted. I would also like to express my appreciation to the chairperson of my thesis com- mittee, Associate Professor Zhuquan Zang, for his encouragement and continual sup- port throughout these years. He has really done a great job in ensuring that my studies were carried out well. I would also like to acknowledge Professor Clem Kuek, the Dean of Research and Development, for his continual support in improving the quality of my research. He ii has always tried to organize research related seminars and talks that are beneficial to improve the quality of the research output. It is also a pleasure to thank my colleagues for their friendly support and friend- ship that blossomed throughout the duration of my studies. They have enriched and broadened my knowledge and perspectives on various aspects of life. I must express my very profound gratitude to my parents, especially my mother, for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis. This accomplishment would not have been possible without them. Last, but not least, I appreciate with gratitude the fee waiver award I received from Faculty of Engineering and Science for granting me an opportunity to conduct MPhil study in Electrical and Computer Engineering, Curtin University Malaysia. Thank you. iii ABSTRACT A nonlinear least squares (NLS) problem commonly arises in nonlinear data-fitting when a nonlinear mathematical model with n unknown parameters is used to fit a set of m observed data with m>n. The best fit to the m observed data is achieved when the residuals between the observed data and its corresponding fitted modeled data are minimized. This is made possible by minimizing an objective function formulated as the sum of squares residual functions of all the m observed data. This procedure is also known as parameter estimation in NLS data-fitting. As a result, the NLS prob- lem is a special class of unconstrained minimization problem and the solution of this minimization problem yields the minimum point which gives the minimal value of the objective function of the NLS problem. Various numerical methods have been developed to solve the NLS problem as un- constrained optimization. These methods can be classified into line search methods or trust region methods. In this thesis, four of the most well-known numerical methods in the NLS literature are considered. The line search methods considered are the steepest descent (SD) method, the Newton’s method and the Gauss-Newton (GN) method while the only trust region method considered is the Levenberg-Marquardt (LM) method. In order to avoid expensive computations of the Hessian matrix in each iteration, the GN and the LM methods use, without justification, a truncated Hessian matrix of the objective function of the NLS problem. However, this truncated Hessian matrix may not be valid especially when the iterations result in large residuals. In addition, iv the computation of the derivatives of the objective function may be prone to analyti- cal mistakes. This is especially true when computing derivatives for high-dimensional NLS problem. To address these issues, numerical differentiation, which uses finite difference approximations, is incorporated into numerical algorithms so that numeri- cal derivatives can be performed by just providing the original objective function of the NLS problem. This saves time and effort while preventing analytical mistakes. Thus, the use of the truncated Hessian matrix can be avoided when developing new numerical methods for solving the NLS problem. With the incorporation of numerical differentiation, all the numerical methods can be implemented in practical problems. The convergence analyses of the numerical methods follow from the Lyapunov function theorem where a sufficient decrease in the objective function is required at every iteration. The Lyapunov function theorem provides a feedback-type analysis which is robust against small numerical errors in the current iteration. If the level sets of the objective function are properly nested, all trajectories will converge to a mini- mum point x provided that the iterations stay within the properly nested region con- taining x. In order to implement the Lyapunov function theorem, all the line search numerical methods perform a backtracking line search so as to ensure a sufficient de- crease of the objective function value of the NLS problem at every iteration. On the other hand, this sufficient decrease requirement of the Lyapunov function theorem is also ensured implicitly in the trust region LM method through the ratio test. When using the MATLAB software to plot the level sets of a two-variable objective function, the level curves near stationary points may not appear in the plot. Hence, a stiff ordinary differential equation (ODE) method, which gives great control to the user, is used as a technique to plot a missing level curve around the stationary points of the objective function through a specific point. This is particularly useful when the objective function has multiple stationary points that are close to each other. The approximate greatest descent (AGD) method has been developed to solve an unconstrained optimization problem. Unlike other methods, the AGD method uses the actual objective function to construct its iterations instead of an approximate linear or v quadratic model. Furthermore, the AGD method is constructed for long-term subopti- mal outcomes to generate the next iterative point on the boundary of the current search region. However, it has not been applied to solve the NLS problem. In addition, a two-phase AGD method (abbreviated as AGDN) is also proposed as a new numerical approach to solve the NLS problem. It consists of two explicitly defined phases with AGD method in Phase-I where the current iterations are far away from the minimum point and then switches to the Newton’s method in Phase-II when the gradient is suf- ficiently small (i.e. near the minimum point). This method is motivated by the fast quadratic convergence rate of the Newton’s method near the minimum point. In order to demonstrate the efficiency, reliability and robustness of all the numerical methods, a standard set of two-variable and multi-variable test problems are selected from Moré et al. (1981) and Adorio (2005) and available in the constrained and un- constrained testing environment, revisited/safe threads (CUTEr/CUTEst) are used to perform numerical experiments. Furthermore, a performance profile is also used as a tool to provide an overall comparison of all the numerical methods in terms of num- ber of iterations and the CPU time used to achieve convergence. When the numerical methods are applied to solve the two-variable and the multi-variable test problems, the numerical results indicate that both the AGD and the AGDN methods have shown en- couraging results in terms of number of iterations and convergence rates as compared to the other methods. For the two-variable test problems, the AGD and the AGDN methods show similar results. However, the outcomes of these methods may differ when they are applied to solve the multi-variable test problems. The results prove that the AGDN method outperforms the AGD method since it has a faster convergence rate with less number of iterations. Nonetheless, the AGDN method may fail to converge if the Hessian matrix is singular near the minimum point. In this case, the AGD method should be used instead. vi ABBREVIATIONS Throughout the report, we use the following abbreviations: NLS: Nonlinear least squares SD: Steepest descent GN: Gauss-Newton LM: Levenberg-Marquardt AGD: Approximate greatest descent AGDN: Two-phase approximate greatest descent TP: Test Problem w.r.t: with respect to vii NOTATIONS Throughout the report, we use the following
Load more

Recommended publications
  • A Time-Stepping Method for Stiff Multibody Dynamics with Contact
    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2002; 55:753–784 (DOI: 10.1002/nme.512) A time-stepping method for sti multibody dynamics with contact and friction‡ Mihai Anitescu1;∗;† and Florian A. Potra2 1Department of Mathematics; University of Pittsburgh; Thackeray 301; Pittsburgh; PA 15213; U.S.A. 2Department of Mathematics; University of Maryland; Baltimore County; Baltimore; MD 21250; U.S.A. SUMMARY We dene a time-stepping procedure to integrate the equations of motion of sti multibody dynamics with contact and friction. The friction and non-interpenetration constraints are modelled by complemen- tarity equations. Stiness is accommodated by a technique motivated by a linearly implicit Euler method. We show that the main subproblem, a linear complementarity problem, is consistent for a suciently small time step h. In addition, we prove that for the most common type of sti forces encountered in rigid body dynamics, where a damping or elastic force is applied between two points of the system, the method is well dened for any time step h. We show that the method is stable in the sti limit, unconditionally with respect to the damping parameters, near the equilibrium points of the springs. The integration step approaches, in the sti limit, the integration step for a system where the sti forces have been replaced by corresponding joint constraints. Simulations for one- and two-dimensional examples demonstrate the stable behaviour of the method. Published in 2002 by John Wiley & Sons, Ltd. KEY WORDS: multibody dynamics; rigid bodies; Coulomb friction; sti methods; linear complemen- tarity problems; linearly implicitly integration 1.
    [Show full text]
  • General Index
    General Index Accelerated convergence of series 172ff. Annealing, method of simulated 398f., 448ff. Accuracy 31f. assessment 459 achievable in minimization 402, 408, 415 for continuous variables 448, 455f. achievable in root finding 357 schedule 449 contrasted with fidelity 843, 851 thermodynamic analogy 448f. CPU different from memory 192 traveling salesman problem 449ff. vs. stability 715, 740, 841, 855 ANSI/ISO C++ standard 3 Accuracy parameters 17, 956f. Antonov-Saleev variant of Sobol’ sequence Acknowledgments xiii 314ff. Adams-Bashford-Moulton method 752 Apple xx Adams’ stopping criterion 377 Macintosh 896 Adaptive integration 133, 146, 714, 719ff., Approximate inverse of matrix 60 729ff., 737f., 742, 748, 753f., 799 Approximation of functions 108f. Adaptive Monte Carlo integration 320ff., by Chebyshev polynomials 197f., 524 323ff. Pade´ approximant 206ff. Addition, multiple precision 917 by rational functions 209ff. Addition theorem, elliptic integrals 266 by wavelets 605f., 793 ADI (alternating direction implicit) method see also Fitting 858, 872f., 917 Arithmetic Adjoint operator 878 arbitrary precision 891, 916ff. Adobe Illustrator xiv, xx floating point 891 Advective equation 837 IEEE standard 289, 893 AGM (arithmetic geometric mean) 917 rounding 892 Airy function 216, 245, 255 Arithmetic coding 891, 912ff. routine for 255f. Arithmetic-geometric mean (AGM) method Aitken’s delta squared process 172 917 Aitken’s interpolation algorithm 111 Array AIX (IBM) 4 C style 20, 941 Algorithms, non-numerical 891ff. centered subarray of 122 Aliasing 506, 581 classes 20ff., 25ff., 941, 950, 953 see also Fourier transform constructors 26, 29 All-poles model 578 index range xi, 21 see also Maximum entropy method (MEM) one-dimensional 20 All-zeros model 578 resize function not assumed 23 see also Periodogram size function 26, 30 Allocation of storage 941 three-dimensional 27 Alternating-direction implicit method (ADI) two-dimensional 20 858, 872f., 917 unit-offset xi, 21 Alternating series 172f.
    [Show full text]
  • AN16-LS16 Abstracts
    AN16-LS16 Abstracts Abstracts are printed as submitted by the authors. Society for Industrial and Applied Mathematics 3600 Market Street, 6th Floor Philadelphia, PA 19104-2688 USA Telephone: +1-215-382-9800 Fax: +1-215-386-7999 Conference E-mail: [email protected] Conference web: www.siam.org/meetings/ Membership and Customer Service: 800-447-7426 (US & Canada) or +1-215-382-9800 (worldwide) 2 2016 SIAM Annual Meeting Table of Contents AN16 Annual Meeting Abstracts ................................................4 LS16 Life Sciences Abstracts ...................................................157 SIAM Presents Since 2008, SIAM has recorded many Invited Lectures, Prize Lectures, and selected Minisymposia from various conferences. These are available by visiting SIAM Presents (http://www.siam.org/meetings/presents.php). 2016 SIAM Annual Meeting 3 AN16 Abstracts 4 AN16 Abstracts SP1 University of Cambridge AWM-SIAM Sonia Kovalevsky Lecture - Bioflu- [email protected] ids of Reproduction: Oscillators, Viscoelastic Net- works and Sticky Situations JP1 From fertilization to birth, successful mammalian repro- Spatio-temporal Dynamics of Childhood Infectious duction relies on interactions of elastic structures with a Disease: Predictability and the Impact of Vaccina- fluid environment. Sperm flagella must move through cer- tion vical mucus to the uterus and into the oviduct, where fertil- ization occurs. In fact, some sperm may adhere to oviduc- Violent epidemics of childhood infections such as measles tal epithelia, and must change their pattern
    [Show full text]
  • General Index to Volumes 1 and 2
    General Index to Volumes 1 and 2 In this index, page numbers 1 through 934 refer to Volume 1, Numerical Recipes in Fortran 77, while page numbers 935 through 1446 refer to Volume 2, Numerical Recipes in Fortran 90. Front matter in Volume 1 is indicated by page numbers in the range 1/i through 1/xxxi, while front matter in Volume 2 is indicated 2/i through 2/xx. Abstract data types 2/xiii, 1030 Alpha AXP 2/xix Accelerated convergence of series 160ff., Alternating-direction implicit method (ADI) 1070 847, 861f., 906 Accuracy 19f. Alternating series 160f., 1070 achievable in minimization 392, 397, 404 Alternative extended Simpson's rule 128 achievable in root ®nding 346f. American National Standards Institute (ANSI) contrasted with ®delity 832, 840 2/x, 2/xiii CPU different from memory 181 Amoeba 403 vs. stability 704, 729, 830, 844 see also Simplex, method of Nelder and Accuracy parameters 1362f. Mead Acknowledgments 1/xvi, 2/ix Ampli®cation factor 828, 830, 832, 840, 845f. Ada 2/x Amplitude error 831 Adams-Bashford-Moulton method 741 Analog-to-digital converter 812, 886 Adams' stopping criterion 366 Analyticity 195 Adaptive integration 123, 135, 703, 708ff., Analyze/factorize/operate package 64, 824 720, 726, 731f., 737, 742ff., 788, 1298ff., Anderson-Darling statistic 621 1303, 1308f. Andrew's sine 697 Monte Carlo 306ff., 1161ff. Annealing, method of simulated 387f., 436ff., Addition, multiple precision 907, 1353 1219ff. Addition theorem, elliptic integrals 255 assessment 447 ADI (alternating direction implicit) method for continuous variables 437, 443ff., 1222 847, 861f., 906 schedule 438 Adjoint operator 867 thermodynamic analogy 437 Adobe Illustrator 1/xvi, 2/xx traveling salesman problem 438ff., 1219ff.
    [Show full text]
  • Advanced Numerical Analysis
    Advanced Numerical Analysis Lecture Notes Prof. Daniel Kressner EPFL / SB / MATHICSE / ANCHP Spring 2015 May 21, 2015 Remarks • Although not necessary, you may find it helpful to complement these lecture notes with additional literature. These lecture notes are based on material from [DB] P. Deuflhard and F. Bornemann. Scientific computing with ordinary differential equations. Texts in Applied Mathematics, 42. Springer, 2002. [E] F. Eisenbrand. Lecture notes on Discrete Optimization, 2013. Available from http://disopt.epfl.ch/Opt2013. [HLW] E. Hairer, Ch. Lubich, and G. Wanner. Geometric numerical integra- tion. Structure-preserving algorithms for ordinary differential equations. Springer Series in Computational Mathematics, 31. Springer, 2010. [HNW] E. Hairer, S. P. Nørsett, and G. Wanner. Solving ordinary differen- tial equations. I. Nonstiff problems. Second edition. Springer Series in Computational Mathematics, 8. Springer, 1993. [HW] E. Hairer and G. Wanner. Solving ordinary differential equations. II. Stiff and differential-algebraic problems. Second revised edition. Springer Series in Computational Mathematics, 14. Springer-Verlag, Berlin, 2010. [K] C. T. Kelley. Iterative Methods for Optimization. SIAM, 1999. [Kr] J. Krieger. Equations Diff´erentielles Ordinaires, 2014. Lecture notes available from http://pde.epfl.ch/page-44573-en.html. [N] Yu. Nesterov. Introductory Lectures on Convex Optimization. A Basic Course, 2004. [NW] J. Nocedal and S. J. Wright. Numerical optimization. Second edi- tion. Springer Series in Operations Research and Financial Engineering. Springer, 2006. [UU] M. Ulbrich and S. Ulbrich. Nichtlineare Optimierung, Birkh¨auserVer- lag, 2012. Please note: None of these books is needed to prepare for the exam. These lecture notes are entirely sufficient. • Sections or paragraphs marked by a ? contain complementary material that is not part of the exam.
    [Show full text]
  • General Index
    General Index Accelerated convergence of series 166ff. schedule 445 Accuracy 28f. thermodynamic analogy 444f. achievable in minimization 398, 404, 410 traveling salesman problem 445ff. achievable in root ®nding 353 ANSI C standard 2f., 14, 25, 930, 941 contrasted with ®delity 841, 849 ANSI macro 17, 930 CPU different from memory 186 Antonov-Saleev variant of Sobol' sequence vs. stability 710, 736, 839, 853 310ff. Acknowledgments xii Apple xvii Adams-Bashford-Moulton method 749 Macintosh 894 Adams' stopping criterion 373 Approximate inverse of matrix 57 Adaptive integration 129, 141, 709, 714ff., Approximation of functions 105f. 725ff., 733f., 737, 744, 749f., 797 by Chebyshev polynomials 191f., 519 Adaptive Monte Carlo integration 316ff., PadÂe approximant 200ff. 319ff. by rational functions 204ff. Addition, multiple precision 916 by wavelets 601f., 791 Addition theorem, elliptic integrals 262 see also Fitting ADI (alternating direction implicit) method Arguments, conversion of data types 24f., 856, 870f., 915 930 Adjoint operator 876 Arithmetic Adobe Illustrator xiii, xvii arbitrary precision 889, 915ff. Advective equation 835 complex 23f., 948ff. AGM (arithmetic geometric mean) 915 ¯oating point 889 Airy function 210, 240, 250 IEEE standard 285, 890f. routine for 250f. rounding 890 Aitken's delta squared process 166 Arithmetic coding 889, 910ff. Aitken's interpolation algorithm 108 Arithmetic-geometric mean (AGM) method Algorithms, non-numerical 889ff. 915 Aliasing 501, 576 Array see also Fourier transform centered subarray of 119 All-poles model 573 how to allocate 19 see also Maximum entropy method (MEM) All-zeros model 573 index range 18 see also Periodogram one-dimensional 18 Allocation of storage 19, 21f., 940ff. relation to C pointer 18 Alternating-direction implicit method (ADI) three-dimensional 23 856, 870f., 915 two-dimensional 20f.
    [Show full text]
  • Evaluating and Generalization of Methods of Cyclic Coordinate, Hooke – Jeeves, and Rosenbrock
    Computational Research 2(3): 31-43, 2014 http://www.hrpub.org DOI: 10.13189/cr.2014.020301 Evaluating and Generalization of Methods of Cyclic Coordinate, Hooke – Jeeves, and Rosenbrock Mohammad Ali Khandan, Mehdi Delkhosh* Department of Mathematics and Computer, Damghan University, Damghan, Iran *Corresponding Author: [email protected] Copyright © 2014 Horizon Research Publishing All rights reserved. Abstract In optimization problems, we are looking for simultaneously). The method of Rosenbrock tests m (m is the best point for our problem. Problems always classify to number of optimized decision variables) orthogonal search unary or n-ary with bound or unbound. In this paper, we directions which are equal to coordinate axes only at the first evaluate three methods, Cyclic Coordinate, Hooke – Jeeves, iteration. Rosenbrock search is a numerical optimization and Rosenbrock on n dimensional space for unbounded algorithm applicable to optimization problems in which the problem and we speak about the advantages and objective function is inexpensive to compute and the disadvantages of these three methods. We offer solutions for derivative either does not exist or cannot be computed some disadvantages and we give generalized algorithms to efficiently. improve the methods. Another type of method, namely dynamic programming, has been used extensively in stand management optimization Keywords Golden Section Method, Cyclic Coordinate [8]. It differs from direct search methods so that, instead of Method, Hooke – Jeeves Method, Rosenbrock Method seeking optimal values for continuous decision variables, it finds the optimal sequence of stand states defined by classified values of several growing stock characteristics (for instance, basal area, mean diameter, and stand age).
    [Show full text]
  • Chapter 1 Introduction
    Chapter Intro duction There is scarcely a mo dern journal whether of engineering economics management mathematics physics or the so cial sciences in which the concept optimization is missing from the sub ject index If one abstracts from all sp ecialist p oints of view the recurring problem is to select a b etter or b est Leibniz eventuallyheintro duced the term optimal alternativefromamonganumb er of p ossible states of aairs However if one were to followthehyp othesis of Leibniz as presented in his Theodicee that our world is the b est of all p ossible worlds one could justiably sink into passive fatalism There would b e nothing to improve or to optimize Biology esp ecially since Darwin has replaced the static world picture of Leibniz time by a dynamic one that of the more or less gradual development of the sp ecies culminat ing in the app earance of man Paleontology is providing an increasingly complete picture of organic evolution Socalled missing links rep eatedly turn out to b e not missing but rather hitherto undiscovered stages of this pro cess Very much older than the recogni tion that man is the result or b etter intermediate state of a meliorization pro cess is the seldomquestioned assumption that he is a p erfect end pro duct the pinnacle of cre ation Furthermore long b efore man conceived of himself as an active participantin the development of things he had unconsciously inuenced this evolution There can b e no doubt that his ability and eorts to make the environment meet his needs raised him ab ove other forms of life
    [Show full text]
  • Peer Methods in Optimal Control
    Peer Methods in Optimal Control Vom Fachbereich Mathematik der Technischen Universität Darmstadt zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Dissertation von Dipl.-Math. Dirk Schröder aus Dieburg Tag der Einreichung: 05.02.2016, Tag der Prüfung: 28.04.2016 Darmstadt 2016 — D 17 1. Gutachten: Prof. Dr. Jens Lang 2. Gutachten: Prof. Dr. Stefan Ulbrich Fachbereich Mathematik AG Numerik und Wissenschaftliches Rechnen Peer Methods in Optimal Control Genehmigte Dissertation von Dipl.-Math. Dirk Schröder aus Dieburg 1. Gutachten: Prof. Dr. Jens Lang 2. Gutachten: Prof. Dr. Stefan Ulbrich Tag der Einreichung: 05.02.2016 Tag der Prüfung: 28.04.2016 Darmstadt 2016 — D 17 Für Helene Zusammenfassung In dieser Arbeit werden implizite und linear implizite Peer-Methoden im Kontext der Optimierung mit gewöhnlichen oder partiellen Differentialgleichungen als Nebenbedin- gungen untersucht. In vielen Anwendungsproblemen, wie dem Kühlen von Glas, einer wandernden Flam- menfront in einem gekühlten Kanal, oder dem Härten von Stahl, können die zugrunde liegenden physikalischen Prozesse durch gewöhnliche oder partielle Differentialglei- chungen modelliert werden. Der Wunsch diese Vorgänge zu optimieren, führt zum Feld der optimalen Steuerung mit Differentialgleichungen. In einem Optimierungsalgorithmus müssen die Nebenbedingungen, also in unserem Fall ein System von gewöhnlichen oder partiellen Differentialgleichungen, mehrmals ausgewertet werden. Daher ist eine effiziente Diskretisierung der Differentialgleichun- gen sehr wichtig. Für gewöhnliche Differentialgleichungen und zur Zeitdiskretisie- rung parabolischer partieller Differentialgleichungen werden gerne Runge-Kutta- und Rosenbrock-Methoden benutzt. Allerdings zeigen diese Methoden bei sehr steifen Pro- blemen und bei der Diskretisierung parabolischer Probleme oft nicht die zu erwartende, klassische Konvergenzordnung. Dieses Phänomen wird Ordnungsreduktion genannt. Eine vielversprechende Alternative sind Peer-Methoden.
    [Show full text]
  • Computational Parameter Selection and Simulation of Complex Sphingolipid Pathway Metabolism
    Computational Parameter Selection and Simulation of Complex Sphingolipid Pathway Metabolism A Thesis Presented to The Academic Faculty by Peter A. Henning In Partial Fulfillment of the Requirements for the Degree Master of Science in Biomedical Engineering Department of Biomedical Engineering Georgia Institute of Technology August 2006 Copyright c 2006 by Peter A. Henning To my family, Dale, Christine, and Sara, I couldn’t have made it through these trying times without you. iii ACKNOWLEDGEMENTS I would like to acknowledge Gautam Goel and Eberhard Voit for their useful insights into past, present, and future of Systems Biology. I would like to thank Dr. Al Merrill and his group for not only providing the sphingolipid experimental data but also for providing expertise in both sphingolipids and mass spectometry. I’d like to thank my fellow researchers Jin Young Hong, John Phan, Pushkar Mukewar, David Stiles, and Paul Tan for answering my at times ridiculous questions on software and various other topics. I would like to also thank my fellow BME classmates for making the semesters fulfilling requirements not only tolerable but at times fun. I’d like to extend my gratitude to Bob Lee and Maggie Cam for their insight into what changes were necessary to make myself a successful researcher and inspiring me to take possession of my graduate degree. I would like to extend my thanks to the BME Department, Georgia Tech, Emory, and NIDDK for this opportunity to be part of cutting edge research. Last but certainly not least, I am grateful for the support from all of my friends.
    [Show full text]
  • Economathematics (M.Sc.) Winter Term 2011/2012 Long Version Date: 09.09.2011
    Economathematics (M.Sc.) Winter Term 2011/2012 Long version Date: 09.09.2011 Department of Economics and Business Engineering Department of Mathematics KIT - University of the State of Baden-Wuerttemberg and www.kit.edu National Research Center of the Helmholtz Association Publishers: Department of Economics and Business Engineering Karlsruhe Institute of Technology (KIT) 76128 Karlsruhe www.wiwi.kit.edu Fakultätfür Mathematik Department of Mathematics Karlsruhe Institute of Technology (KIT) 76128 Karlsruhe www.math.kit.edu Contact: [email protected] [email protected] 2 CONTENTS CONTENTS For informational use only. For legally binding information please refer to the german version of the handbook. Contents 1 Studyplan 9 2 Helpful information 23 3 Actual Changes 25 4 Modules 27 4.1 Modules of Mathematics .......................................... 27 Riemannian Geometry- MATHMWAG04.................................. 27 Algebra- MATHMWAG05.......................................... 28 Discrete Geometry- MATHMWAG06.................................... 29 Convex Geometry- MATHMWAG07..................................... 30 Geometric Measure Theory- MATHMWAG08............................... 31 Algebraic Number Theory- MATHMWAG09................................ 32 Algebraic Geometry- MATHMWAG10................................... 33 Geometry of Schemes- MATHMWAG11.................................. 34 Geometric Group Theory- MATHMWAG12................................. 35 Lie Groups and Lie Algebras- MATHMWAG13............................... 36 Metric
    [Show full text]
  • 1985 Mathematical Sciences Professional Directory
    Notices of the American Mathematical Society June 1985, Issue 241 Volume 32, Number 3, Pages 337- 456 Providence, Rhode Island USA ISSN 0002-9920 Calendar of AMS Meetings THIS CALENDAR lists all meetings which have been approved by the Council prior to the date this issue of the Notices was sent to the press. The summer and annual meetings are joint meetings of the Mathematical Association of America and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to change: this is particularly true of meetings to which no numbers have yet been assigned. Programs of the meetings will appear in the issues indicated below. First and supplementary announcements of the meetings will have appeared in earlier issues. ABSTRACTS OF PAPERS presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meeting. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the office of the Society. Abstracts must be accompanied by the $15 processing charge. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Society in Providence. Rhode Island. on or before the deadline given below for the meeting. Note that the deadline for abstracts for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information consult the meeting announcements and the list of organizers of special sessions.
    [Show full text]